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arXiv:2207.01827 · Philosophical Transactions of the Royal Society A Mathematical Physical and Engineering Sciences

Quantum annealing: an overview

Atanu Rajak, Sei Suzuki, Amit Dutta, Bikas K. Chakrabarti

Published Dec 4, 2022• 170 citations
arXivHTMLPDF

Abstract

In this review, after providing the basic physical concept behind quantum annealing (or adiabatic quantum computation), we present an overview of some recent theoretical as well as experimental developments pointing to the issues which are still debated. With a brief discussion on the fundamental ideas of continuous and discontinuous quantum phase transitions, we discuss the Kibble-Zurek scaling of defect generation following a ramping of a quantum many body system across a quantum critical point. In the process, we discuss associated models, both pure and disordered, and shed light on implementations and some recent applications of the quantum annealing protocols. Furthermore, we discuss the effect of environmental coupling on quantum annealing. Some possible ways to speed up the annealing protocol in closed systems are elaborated upon: we especially focus on the recipes to avoid discontinuous quantum phase transitions occurring in some models where energy gaps vanish exponentially with the system size. This article is part of the theme issue 'Quantum annealing and computation: challenges and perspectives'.

DOI
10.1098/rsta.2021.0417

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Concluding Comments

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In this book, we have discussed quantum phase transitions in transverse field Ising and XY models in one and higher dimensions, particularly in the context of quantum information processing and non-equilibrium dynamics. We have illustrated how these models relate to actual quantum annealing protocols in physical systems and their connection to efficient quantum computation. Quantum phase transitions for pure models as well as for models with random interactions or random fields have also been considered. These models were introduced in the early 1960's in the context of order-disorder ferroelectric systems. This book focuses on the salient issues for which these models continue to be important and interesting even fifty years after their first appearance. Our starting point involved generic theories of behavior of information theoretic measures, which were then illustrated using the transverse field models. Related models, e.g. those which are exactly soluble using Jordan-Wigner transformations and those exhibiting topological quantum phase transitions, were presented in the book at some length. These models are relevant to low-dimensional condensed matter physics as well as to quantum information and quantum dynamical studies. On the theoretical side, the integrability of the transverse field Ising/ XY models in one dimension has provided an ideal testing ground for field theoretical and information theoretical studies. Numerous predictions of theoretical studies have been verified experimentally in recent years. [180]. The one-dimensional version of these models have been extremely useful in studying information theoretic measures like concurrence, entanglement entropy, fidelity and fidelity susceptibility, and also the scaling of the defect density generated by quantum critical and multicritical quenches, namely, the Kibble–Zurek scaling. These models have played a crucial role in the development of quantum annealing techniques and adiabatic quantum algorithms. The equilibration or thermalization following quantum quenches have been studied using variants of transverse field models. An advantageous feature of transverse field Ising models is the quantum-classical mapping which renders these models ideally suited for quantum Monte Carlo studies for d > 1.

Cambridge University Press eBooks•Jan 28, 2015•0 citations
In this review, after providing the basic physical concept behind quantum annealing (or adiabatic quantum computation), we present an overview of some recent theoretical as well as experimental developments pointing to the issues which are still debated. With a brief discussion on the fundamental ideas of continuous and discontinuous quantum phase transitions, we discuss the Kibble-Zurek scaling of defect generation following a ramping of a quantum many body system across a quantum critical point. In the process, we discuss associated models, both pure and disordered, and shed light on implementations and some recent applications of the quantum annealing protocols. Furthermore, we discuss the effect of environmental coupling on quantum annealing. Some possible ways to speed up the annealing protocol in closed systems are elaborated upon: we especially focus on the recipes to avoid discontinuous quantum phase transitions occurring in some models where energy gaps vanish exponentially with the system size. This article is part of the theme issue 'Quantum annealing and computation: challenges and perspectives'.
Quantum annealing is a computing paradigm that has the ambitious goal of efficiently solving large-scale combinatorial optimization problems of practical importance. However, many challenges have yet to be overcome before this goal can be reached. This perspectives article first gives a brief introduction to the concept of quantum annealing, and then highlights new pathways that may clear the way towards feasible and large scale quantum annealing. Moreover, since this field of research is to a strong degree driven by a synergy between experiment and theory, we discuss both in this work. An important focus in this article is on future perspectives, which complements other review articles, and which we hope will motivate further research.
We study errors of quantum annealing and simulated annealing to highlight better performance of quantum annealing over simulated annealing. Quantum annealing and simulated annealing perform optimization through a quantum adiabatic evolution and a quasi-static evolution respectively. In both methods, dynamics across a phase transition plays a crucial role. The Kibble-Zurek mechanism is known as an underlying physics of defect formation during a time-evolution across a phase transition. We apply an argument for the Kibble-Zurek mechanism to the error generation of quantum annealing and simulated annealing. We show that, for the disordered Ising chain, the kink density and residual energy per spin of quantum annealing decay as (ln τ ) −2 and (ln τ ) −4 respectively, whereas those of simulated annealing decay as (ln τ ) −1 and (ln τ ) −2 with the annealing rate −1/ τ . These results imply better performance of quantum annealing. We also develop our theory for a two-dimensional spin-glass model.
We review quantum phase transitions of spin systems in transverse magnetic fields taking the examples of the spin-1/2 Ising and XY models in a transverse field. Beginning with an overview of quantum phase transitions, we introduce a number of model Hamiltonians. We provide exact solutions in one spatial dimension connecting them to conformal field theoretical studies. We also discuss Kitaev models and some other exactly solvable spin systems. Studies of quantum phase transitions in the presence of quenched randomness and with frustrating interactions are presented in detail. We discuss novel phenomena like Griffiths-McCoy singularities. We then turn to more recent topics like information theoretic measures of the quantum phase transitions in these models such as concurrence, entanglement entropy, quantum discord and quantum fidelity. We then focus on non-equilibrium dynamics of a variety of transverse field systems across quantum critical points and lines. After mentioning rapid quenching studies, we dwell on slow dynamics and discuss the Kibble-Zurek scaling for the defect density following a quench across critical points and its modifications for quenching across critical lines, gapless regions and multicritical points. Topics like the role of different quenching schemes, local quenching, quenching of models with random interactions and quenching of a spin chain coupled to a heat bath are touched upon. The connection between non-equilibrium dynamics and quantum information theoretic measures is presented at some length. We indicate the connection between Kibble-Zurek scaling and adiabatic evolution of a state as well as the application of adiabatic dynamics as a tool of a quantum optimization technique known as quantum annealing. The final section is dedicated to a detailed discussion on recent experimental studies of transverse Ising-like systems.
Demonstrating genuine many-body quantum coherence in large-scale quantum processors remains a central challenge for near-term quantum technologies. Recent experiments on D-Wave quantum annealers have investigated quenches of Ising chains and observed defect densities that show Kibble-Zurek scaling, consistent with coherent quantum dynamics. However, identical scaling can arise from classical or thermal processes. Here we propose the use of many-body coherent oscillations (MBCO) as a diagnostic for the identification of system-wide coherence in analog quantum simulators. Solving the time-dependent Schrodinger equation, we show that quenches of a staggered one-dimensional Ising chain across a quantum critical point produce oscillatory signatures in defect observables. We implement this model on the D-Wave Advantage quantum annealer. Using fast-anneal protocols, we find that, although defect densities follow Kibble-Zurek scaling, the expected oscillatory behavior is absent. We demonstrate that static disorder associated with individual qubits is not likely responsible for the absence of MBCO. Modest modifications to annealing schedules can dramatically enhance oscillation visibility. This work gives a general roadmap for the search for quantum coherence in noisy, large-scale quantum platforms.
In this book, we have discussed quantum phase transitions in transverse field Ising and XY models in one and higher dimensions, particularly in the context of quantum information processing and non-equilibrium dynamics. We have illustrated how these models relate to actual quantum annealing protocols in physical systems and their connection to efficient quantum computation. Quantum phase transitions for pure models as well as for models with random interactions or random fields have also been considered. These models were introduced in the early 1960's in the context of order-disorder ferroelectric systems. This book focuses on the salient issues for which these models continue to be important and interesting even fifty years after their first appearance. Our starting point involved generic theories of behavior of information theoretic measures, which were then illustrated using the transverse field models. Related models, e.g. those which are exactly soluble using Jordan-Wigner transformations and those exhibiting topological quantum phase transitions, were presented in the book at some length. These models are relevant to low-dimensional condensed matter physics as well as to quantum information and quantum dynamical studies. On the theoretical side, the integrability of the transverse field Ising/ XY models in one dimension has provided an ideal testing ground for field theoretical and information theoretical studies. Numerous predictions of theoretical studies have been verified experimentally in recent years. [180]. The one-dimensional version of these models have been extremely useful in studying information theoretic measures like concurrence, entanglement entropy, fidelity and fidelity susceptibility, and also the scaling of the defect density generated by quantum critical and multicritical quenches, namely, the Kibble–Zurek scaling. These models have played a crucial role in the development of quantum annealing techniques and adiabatic quantum algorithms. The equilibration or thermalization following quantum quenches have been studied using variants of transverse field models. An advantageous feature of transverse field Ising models is the quantum-classical mapping which renders these models ideally suited for quantum Monte Carlo studies for d > 1.